Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

==== Your draft applied the extended linear limit LLL or passed to a limit of Picard iterates and then asserted the energy equality holds for the limit. This step is invalid unless you have one of the two things above: ====
* Either you have shown the Picard iterates converge strongly in the norms that control the energy (i.e. strong Lt∞Lx2L^\infty_t L^2_xLt∞Lx2 convergence and strong Lt2Hx1L^2_t H^1_xLt2Hx1 convergence), so you can pass the equality term-by-term;
* Or you have proven that the linear limit operator LLL has the continuity properties required to carry the nonlinear quadratic forms and integrals through — something you have not established.

Without that, you can at best conclude the energy inequality (via weak lower-semicontinuity) not the exact conservation/dissipation equality. And the energy inequality alone is not enough to rule out finite-time blow-up of classical norms (it controls only L2L^2L2 and the dissipation integral).

Hence the claim “no finite-time blowup” is unsupported: you either must prove additional a-priori bounds in supercritical norms or show the limit is classical (strong convergence) so you can upgrade to equality and then bootstrap regularity — neither has been done.